How much should we trust R2 and adjusted R2: evidence from regressions in top economics journals and Monte Carlo simulations

R2 and adjusted R2 may exaggerate a model’s true ability to predict the dependent variable in the presence of overfitting, whereas leave-one-out R2 (LOOR2) is robust to overfitting. We demonstrate this by replicating 279 regressions from 100 papers in top economics journals, where the median increases of R2 and adjusted R2 over LOOR2 reach 40.2% and 21.4% respectively. The inflation of test errors over training errors increases with the severity of overfitting as measured by the number of regressors and nonlinear terms, and the presence of outliers, but decreases with the sample size. These results are further validated by Monte Carlo simulations.当存在过拟合（overfitting）时，决定系数（R²）与调整后决定系数（adjusted R²）可能会高估模型对因变量的真实预测能力，而留一法决定系数（leave-one-out R²，LOOR2）则对过拟合具备鲁棒性。我们通过复现经济学顶刊100篇论文中的279项回归分析验证了这一结论，结果显示，决定系数与调整后决定系数相较于留一法决定系数的中位数增幅分别可达40.2%与21.4%。测试误差相较于训练误差的膨胀程度，会随着过拟合程度的加深而加剧——过拟合程度可通过回归元数量、非线性项数量以及异常值的存在情况进行衡量，但会随样本量的增大而减弱。上述结果已通过蒙特卡洛模拟（Monte Carlo simulations）得到进一步验证。